A secant of a one is a line that intersects a circle in ~ two unique points. Secant is obtained from the Latin word secare which way to cut. It can likewise be understood as the extension of the chord that a circle the goes exterior the circle.

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1.What is Secant that a Circle?
2.Secant the a one Examples
3.Secant Theorems
4.Tangent and Secant that a circle
5.FAQs on Secant of a Circle

Secant the a circle is the line that cuts throughout the one intersecting the circle in ~ 2 distinctive points. In the one below, PQ is the secant line that cuts the one at 2 points A and also B.

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Difference between a Chord and also a Secant

When a secant line cut the one at 2 points, we gain a chord in ~ the 2 points of intersection. The chord of a circle is a heat segment who endpoints lie on the circular arc. In the circle shown above, abdominal is the chord i m sorry is a section of the secant line QP. In other words, a chord is a heat segment joining two points top top the circumference of the circle, and if this chord is extended on both political parties it becomes the secant. The secant line that passes through the facility of the one produces the diameter. Therefore a secant line determines the chord or diameter in a circle.


Secant that a circle Examples


In genuine life, we come throughout a secant of a one in many places, where the circles or curves are involved. Because that example, in the building and construction of bent bridges, in finding the distance in between the orbiting moon and also the different locations on earth, and also so on. Over there are many interesting properties of secants that aid in obscure geometric constructions. Over there are many circle theorems based upon the secants and also the intersecting secants that a circle.


Secant Theorems


The intersecting secants theorem says that once two secants crossing at an exterior point, the product of the one whole secant segment and its exterior segment is same to the product that the other whole secant segment and its exterior segment. This is also known together the secant organize or the secant power theorem.

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In the figure displayed above, we discover that abdominal and AC room the 2 secant segment intersecting at suggest A. Ad is the external secant segment that the whole secant segment AB, and AE is the outside secant segment that AC. Thus, according to the theorem, us have ab × advertisement = AC × AE

Secants and Angle Measures

Two secants can intersect within or exterior a circle. In the circles shown below, we uncover that the intersecting secants inside and outside create angles x and also y at the clues of intersection, respectively. In the an initial circle, the secants crossing inside the circle and significant arc ad and young arc BD space intercepted by the secants. In the 2nd circle, the secants intersect external the circle and also the significant arc PT and also minor arc QS room intercepted by the secants.

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There space two theorems based upon this residential or commercial property of secants. As per the theorem, us have:

The angle formed by the 2 secants that intersect inside the circle is fifty percent the sum of the intercepted arcs.The angle formed by the 2 secants which intersect external the circle is half the distinction of the intercepted arcs.

Tangent and also Secant the a Circle


Tangents and secants space the lines that cut the circle and extend in both direction infinitely. The main difference between them is that a secant cut the circle at two points, whereas, a tangent cuts the circle in ~ one point. The tangent is perpendicular to the radius at the suggest of the tangency.

Tangent Secant Theorem

According come the tangent secant theorem, if a secant and also a tangent are attracted to a circle native a usual exterior point, climate the product of the length of the entirety secant segment and also its exterior secant segment is same to the square that the size of the tangent segment.

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Observe the number given above to check out that:

The secant AC and the tangent CD are drawn from the same exterior point. Secant segments are abdominal muscle (interior) and BC(exterior). The product that the secant and its exterior segment is same to the square that the tangent segment. AC × BC = CD2The angle subtended through the tangent and also the secant in ~ the exterior is half the distinction of the major arc and also the young arc intercepted through them. \(\alpha = \dfrac12< \overline\rm AD - \overline\rm BD>\)

Check out below for a couple of interesting topics pertained to secant of a circle:

Important Notes

Here is a list of a few points that must be remembered if studying about the secant the a circle:

A secant of a one is a line that connects two distinct points top top a curve.The intersecting secants theorem states that if we attract two secant lines native an exterior allude of a circle, the product the one secant and also its outside segment is same to the product of the various other secant and its external segment.The secant-tangent preeminence states that as soon as a secant line and a tangent line are attracted both from a common exterior point, the product that the secant and its external segment is equal to the square that the tangent segment.

Example 1: Find the size of the tangent segment abdominal given the measure of the secant segments ad and DC together 5 units and 15 systems respectively.

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Solution:

The whole secant segment of the circle is AC and also it intersects in ~ D and also C.

AD + DC = AC

The secant the the circle procedures 15 + 5 = 20 units.

According to the secant tangent rule, we recognize that: (the entirety secant segment × the exterior secant segment) = square that the tangent. Here, AC is the totality secant segment, advertisement is the exterior secant segment, abdominal muscle is the tangent.

AC × AD= AB2

20 × 5 = AB2

100 = AB2

AB = 10 units.

Therefore, the length of the tangent abdominal muscle = 10 units.

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Example 2: Find the absent angle x° utilizing the intersecting secants organize of a circle, provided arc QS = 75° and arc PR= x°.

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Solution:

Using the secant the a one formula (intersecting secants theorem), we recognize that the angle formed in between 2 secants = (1/2) (major arc + minor arc)